Question

How far must a spring with a spring constant of 85 N>m be stretched to store 0.22 J of potential energy?

Short answer

The spring must be stretched approximately 0.072 m or 7.2 cm.

Step-by-step solution

Understanding the Spring Potential Energy Formula

The potential energy stored in a spring can be calculated using the formula: \[ PE = \frac{1}{2}kx^2 \]where \( PE \) is the potential energy, \( k \) is the spring constant, and \( x \) is the displacement or stretch of the spring from its equilibrium position.

Rearranging the Formula to Solve for x

To find the amount of stretch \( x \), we need to rearrange the formula:\[ PE = \frac{1}{2}kx^2 \]Solving for \( x \) gives:\[ x = \sqrt{\frac{2PE}{k}} \]

Substituting the Given Values

Substitute \( PE = 0.22 \, \text{J} \) and \( k = 85 \, \text{N/m} \) into the rearranged formula:\[ x = \sqrt{\frac{2 \times 0.22}{85}} \]

Calculating the Stretch x

Perform the calculation:\[ x = \sqrt{\frac{0.44}{85}} \]\[ x = \sqrt{0.005176} \]\[ x \approx 0.072 \, \text{m} \] or \( 7.2 \, \text{cm} \).

Key concepts

Spring Constant

The spring constant, often symbolized as \( k \), is a measure of a spring's stiffness. It essentially tells us how much force is needed to stretch or compress the spring by a certain length.
The unit of the spring constant is Newtons per meter (N/m) in the SI system.When a spring has a high spring constant, it means it's quite stiff and difficult to stretch. Conversely, a spring with a low spring constant is more easily stretched or compressed.
This concept is vital in physics as it helps determine how a spring behaves under different forces.Springs with different constants are used in various real-world applications, from mattress springs to car suspensions.
Understanding the spring constant is essential when calculating how systems with springs work effectively.

Displacement

In the context of springs, displacement refers to how much a spring is stretched or compressed from its original, or equilibrium, position.Displacement is generally denoted by \( x \), and it is measured in meters (m) in the metric system.
This measure is crucial because it directly affects the potential energy stored in the spring.When calculating the displacement of a spring for a given potential energy, we need to consider both the spring constant and the energy in the system.
Displacement can be a positive or negative value, depending on whether the spring is stretched or compressed.

Energy Calculation

The energy calculation for a spring comes into play when determining the potential energy stored as a result of stretching or compressing the spring.The formula to calculate this potential energy is: \[ PE = \frac{1}{2}kx^2 \]where \( PE \) is the potential energy, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.This formula indicates that potential energy is proportional to both the square of the displacement and the spring constant.
Therefore, increasing the displacement or using a stiffer spring will result in higher potential energy stored in the spring.Understanding how to perform these calculations allows for predicting and manipulating the behavior of spring systems in practical scenarios.

Equilibrium Position

The equilibrium position of a spring refers to the position where the spring naturally rests without any forces acting upon it. When a spring is at its equilibrium position, its potential energy is zero, as there is no external force causing it to stretch or compress.
In this state, the forces on the spring are perfectly balanced. Any displacement from this position results in a change in potential energy and often involves either stretching or compressing the spring. Understanding the equilibrium position is vital when assessing how forces and displacements impact the potential energy in a spring.
This concept is also fundamental in designing mechanical systems that rely on springs for balance and stability.